Simplify (6i)/(9-9i)

$\frac{6 i}{9 - 9 i}$

Cancel the common factor of $6$ and $9 - 9 i$ .

Factor $3$ out of $6 i$ .

$\frac{3 (2 i)}{9 - 9 i}$

Cancel the common factors.

Factor $3$ out of $9$ .

$\frac{3 (2 i)}{3 (3) - 9 i}$

Factor $3$ out of $- 9 i$ .

$\frac{3 (2 i)}{3 (3) + 3 (- 3 i)}$

Factor $3$ out of $3 (3) + 3 (- 3 i)$ .

$\frac{3 (2 i)}{3 (3 - 3 i)}$

Cancel the common factor.

$\frac{3 (2 i)}{3 (3 - 3 i)}$

Rewrite the expression.

$\frac{2 i}{3 - 3 i}$

Multiply the numerator and denominator of $\frac{2 i}{3 - 3 i}$ by the conjugate of $3 - 3 i$ to make the denominator real.

$\frac{2 i}{3 - 3 i} \cdot \frac{3 + 3 i}{3 + 3 i}$

Multiply.

Combine.

$\frac{2 i (3 + 3 i)}{(3 - 3 i) (3 + 3 i)}$

Simplify the numerator.

Apply the distributive property.

$\frac{2 i \cdot 3 + 2 i (3 i)}{(3 - 3 i) (3 + 3 i)}$

Multiply $3$ by $2$ .

$\frac{6 i + 2 i (3 i)}{(3 - 3 i) (3 + 3 i)}$

Multiply $2 i (3 i)$ .

Multiply $3$ by $2$ .

$\frac{6 i + 6 i i}{(3 - 3 i) (3 + 3 i)}$

Raise $i$ to the power of $1$ .

$\frac{6 i + 6 (i^{1} i)}{(3 - 3 i) (3 + 3 i)}$

Raise $i$ to the power of $1$ .

$\frac{6 i + 6 (i^{1} i^{1})}{(3 - 3 i) (3 + 3 i)}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 i + 6 i^{1 + 1}}{(3 - 3 i) (3 + 3 i)}$

Add $1$ and $1$ .

$\frac{6 i + 6 i^{2}}{(3 - 3 i) (3 + 3 i)}$

Simplify each term.

Rewrite $i^{2}$ as $- 1$ .

$\frac{6 i + 6 \cdot - 1}{(3 - 3 i) (3 + 3 i)}$

Multiply $6$ by $- 1$ .

$\frac{6 i - 6}{(3 - 3 i) (3 + 3 i)}$

Reorder $6 i$ and $- 6$ .

$\frac{- 6 + 6 i}{(3 - 3 i) (3 + 3 i)}$

Simplify the denominator.

Expand $(3 - 3 i) (3 + 3 i)$ using the FOIL Method.

Apply the distributive property.

$\frac{- 6 + 6 i}{3 (3 + 3 i) - 3 i (3 + 3 i)}$

Apply the distributive property.

$\frac{- 6 + 6 i}{3 \cdot 3 + 3 (3 i) - 3 i (3 + 3 i)}$

Apply the distributive property.

$\frac{- 6 + 6 i}{3 \cdot 3 + 3 (3 i) - 3 i \cdot 3 - 3 i (3 i)}$

Simplify.

Multiply $3$ by $3$ .

$\frac{- 6 + 6 i}{9 + 3 (3 i) - 3 i \cdot 3 - 3 i (3 i)}$

Multiply $3$ by $3$ .

$\frac{- 6 + 6 i}{9 + 9 i - 3 i \cdot 3 - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{- 6 + 6 i}{9 + 9 i - 9 i - 3 i (3 i)}$

Multiply $3$ by $- 3$ .

$\frac{- 6 + 6 i}{9 + 9 i - 9 i - 9 i i}$

Raise $i$ to the power of $1$ .

$\frac{- 6 + 6 i}{9 + 9 i - 9 i - 9 (i^{1} i)}$

Raise $i$ to the power of $1$ .

$\frac{- 6 + 6 i}{9 + 9 i - 9 i - 9 (i^{1} i^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{- 6 + 6 i}{9 + 9 i - 9 i - 9 i^{1 + 1}}$

Add $1$ and $1$ .

$\frac{- 6 + 6 i}{9 + 9 i - 9 i - 9 i^{2}}$

Subtract $9 i$ from $9 i$ .

$\frac{- 6 + 6 i}{9 + 0 - 9 i^{2}}$

Add $9$ and $0$ .

$\frac{- 6 + 6 i}{9 - 9 i^{2}}$

Simplify each term.

Rewrite $i^{2}$ as $- 1$ .

$\frac{- 6 + 6 i}{9 - 9 \cdot - 1}$

Multiply $- 9$ by $- 1$ .

$\frac{- 6 + 6 i}{9 + 9}$

Add $9$ and $9$ .

$\frac{- 6 + 6 i}{18}$

Cancel the common factor of $- 6 + 6 i$ and $18$ .

Factor $6$ out of $- 6$ .

$\frac{6 \cdot - 1 + 6 i}{18}$

Factor $6$ out of $6 i$ .

$\frac{6 \cdot - 1 + 6 (i)}{18}$

Factor $6$ out of $6 \cdot - 1 + 6 (i)$ .

$\frac{6 \cdot (- 1 + i)}{18}$

Cancel the common factors.

Factor $6$ out of $18$ .

$\frac{6 \cdot (- 1 + i)}{6 (3)}$

Cancel the common factor.

$\frac{6 \cdot (- 1 + i)}{6 \cdot 3}$

Rewrite the expression.

$\frac{- 1 + i}{3}$

Split the fraction $\frac{- 1 + i}{3}$ into two fractions.

$\frac{- 1}{3} + \frac{i}{3}$

Move the negative in front of the fraction.

$- \frac{1}{3} + \frac{i}{3}$

Simplify (6i)/(9-9i)

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